How do I find my serial number if I don't have the packaging and my phone is locked in recovery mode

The last section of the following gives a method (if you've ever backed up to iTunes): Find the serial number and other information for your iPhone, iPad, and iPod touch - Apple Support

Another method is to go to https://supportprofile.apple.com/ if you have registered it there.

If you have never backed up on iTunes then it doesn't matter, but if you are in doubt or think you may have backed up a long time ago and possibly forgotten, you can do as instructed in the link I posted above. You do NOT connect your phone to the computer to do the following from that link, so there is no chance of it having any effect on your phone.

If you can use iTunes on your computer but don't have your device:

1. Go to iTunes > Preferences (Mac) or Edit > Preferences (PC).
2. Click the Devices tab.
3. Hold your mouse over a backup to display the serial number for that device. You'll also see the IMEI/MEID and CDN for your iPhone or iPad (cellular model).